Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...
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2answers
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Reason for existence of 'swapping' elementary matrix operation
In my Hoffman & Kunze book on Linear Algebra, we define 3 elementary row operations, the first of which is the swapping of two rows.
I'm wondering why we need to even have such an elementary ...
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1answer
83 views
Defining a mathematical doodad [closed]
When you define something in mathematics, can you do no appealing to what it does?
For example, I can define the a function by its outcome. f(x) = 2x is something that doubles x. But, I wouldn't be ...
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1answer
176 views
Geometric Definitions: What is a straight line? What is a circle?
What is a straight line? I need a geometric definition of it. The equation of a straight line is known to me.I am saying about a straight line of 2D plane.
What is a circle? I need a geometric ...
5
votes
4answers
390 views
Is the Dirac Delta “Function” really a function?
I am given to understand that the Dirac delta function is strictly not a function in the conventional sense and it is a "functional or a distribution".
The part which I can not understand why the ...
2
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2answers
82 views
The problem of bound variables in mathematical definitions
I was reading Paul Bernays’ Axiomatic Set Theory recently; in the book, Bernays gives the following definition of ‘ordinal number’.
\begin{align}
\text{On}(\alpha) \stackrel{\text{def}}{\iff}
...
3
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1answer
71 views
Are distinctions in definitions of “finite” material in, eg, topology or measure theory?
There are several definitions of "finite", like Dedekind's and Tarski's (Thanks to A.K. for point out the latter - first time I've heard of it): From the Wikipedia entry on Finite Set:
(Richard ...
3
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3answers
52 views
Definition of a tensor field
Could anybody explain to me the following:
If $$T_{ij}=\nabla_i V_j-\nabla_j V_i$$
where $V_i$ is a covector field and $\nabla_i$ is the covariant derivative,
then $T_{ij}$ is a tensor field.
...
5
votes
3answers
79 views
Understanding a definition of radial
A space $X$ is called radial if, for any $A \subset X$ and any $x \in cl(A)$, there is a transfinite sequence $s=\{a_\alpha: \alpha \in \kappa\} \subset A$ which converges to $x$. What's meaning of ...
2
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1answer
63 views
Can we apply a successor operation in Peano's arithmetic infinitely many times, is the successor well defined?
If we look at the axioms of Peano arithmetic, e.g. http://mathworld.wolfram.com/PeanosAxioms.html, they contain an axiom:
If $a$ is a number, the successor of $a$ is a number.
However, the axioms do ...
2
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2answers
123 views
What does lg x mean? is it $\log_2 x$ or $\log_{10} x$ in binary trees
I'm a bit confused, $\log_{10} x = \log x $ right? I believe I've read somewhere that $\log_{2} x = lg x$ but some people say lg = $\log$.
So what does lg really stand for? specifically when talking ...
2
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1answer
81 views
What does compact cover mean?
I am reading a difinition of Lindelof $\Sigma$ space. It talked about compact cover. As the title explains, what does compact cover mean? It means every member of the cover is compact?
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1answer
129 views
What is an algebra?
Is an algebra or 'a algebra' the same thing as an algebraic structure?
Or does it have a different meaning?
Thanks
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1answer
32 views
A Quadratic Maximum?
What does the following mean?
Context: Laplace integrals
Consider the integral $$\int_a^b f(t) \exp(x\phi(t))\,\,\,dt$$ where $\phi(t) $ achieves its maximum at some $t=c\in (a,b)$. Assume that ...
3
votes
1answer
76 views
A question on linear ordered space
A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$. And we know every space contains a dense left-separated subspace.
My question ...
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2answers
123 views
Definition of a basis for a particular topological space
I'm currently looking at Lemma 13.2 in Munkres' Topology. It states the following: Given a collection $C$ of open sets of a topological space $X$ such that for each open set $U$ of $X$ and each $x$ in ...
2
votes
3answers
96 views
What is the definition of first/last element in a poset?
I have read the terms first element/last elements in the context of
a basic course in set theory.
When I learned about posets I didn't encounter those terms. I tried
looking up the definitions but I ...
1
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1answer
42 views
Use of the term “normal section” in a theorem of Maria Lucido.
Prop. 3 in this paper (p.135) states
Let $G$ be a solvable group with $\text{diam}\Gamma(G)=4$. Then either $l_F(G)\leq 3$ or $l_F(G)=4$ and $G$ has a normal section isomorphic to $H$.
($H$ is ...
1
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1answer
74 views
Definition for series with negative index and order of taking limits
I have thee questions and they seem all related to me and every number i say is complex number below.
My definition for $\sum_{k=0}^n=a_k$ is by induction. That is, given finite set of numbers ...
2
votes
0answers
38 views
Can a sample space depend on the parameter to be estimated (example: # of cabs in a city)
In our intro statistic lecture the following we said that the following components made up an estimation problem
an at most countable space $\mathcal{X}$ of all possible samples we can observe
a ...
2
votes
4answers
339 views
Definition for Covariant Derivative
What is simple definition of the covariant derivative that looks like the definition of the derivative of a function in calculus?
definition of the derivative of a function in calculus is:
$$\frac ...
5
votes
2answers
199 views
Why does the condition of a function being differentiable always require an open domain?
Going through Spivak's Calculus on Manifolds and in his definition of a differentiable function from a subset $A$ of $\mathbb{R}^n$ to $\mathbb{R}^m$, $f$ is said to be differentiable if it can be ...
5
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4answers
278 views
Can open sets be an open cover, for itself?
I have Baby Rudin's book with me and it clearly defines a cover to be open. In a followup, it defines a set $K$ to be compact if every open cover of $K$ contains a finite subcover.
And the rest I ...
2
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2answers
68 views
Is the definition if mutual independence too strong?
Let $A_1,\ldots,A_n$ be $n$ events in a discrete probability space. Can someone give me an example such that $$P(A_1 \cap \cdots \cap A_n)=P(A_1)\cdot \ldots \cdot P(A_n)$$ holds, but such that there ...
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4answers
147 views
Is a linear combination linearly independent?
I am a bit confused... Linear combination means
$$F(X)=af(x_1)+bf(x_2) + \cdots$$
and linearly independent means
$$af(x_1)+bf(x_2) + \cdots=0$$
where $a=b=\cdots=0$
My question: is a linear ...
2
votes
1answer
134 views
Base (topology) with closed intervals
I am curious why it's a problem to define a base using closed sets?
For example, my book uses the definition under "Constructing Topologies from Bases" as specified at ...
6
votes
2answers
174 views
Understanding induced representations
Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these.
My question is: ...
2
votes
2answers
59 views
A simple notation question on grad and Lp norm
What does this notation
$||\nabla u|| _p$ mean? I would like an exact definition.
Also I have seen $|\nabla u|_1$. Does this mean the $\sum_i|\partial_i u|$?
2
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0answers
69 views
Graph Theory: Help with a definition
I need some help to see if the definition I found of Cycles and Cycles Decomposition is right, here it is:
A graph is a Cycle if it is isomorph to another graph $G=(V,E)$ with the following ...
4
votes
1answer
135 views
What is “algebra” in $\sigma$-algebra (or “field” in $\sigma$-field)?
I know that $\sigma$ in $\sigma$-algebra stands for the closure under countable union property. What about "algebra"? Surely it cannot algebra over a field or a ring as defined in algebra textbooks. ...
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5answers
155 views
Is $\{-2,2\}$ a group under $a\star b=\max\{a,b\}$?
Lets say $G={-2,2}$ and $a*b=\text{max}\{a,b\}$.
I need to check if this is a group
and if it does than is it abelian or not and finite or not.
Well... first, I'm not sure if this is a group. for ...
2
votes
1answer
108 views
bipartite graph - sufficient and necessary conditions
Sorry for a silly question, I got confused with the definition of bipartite graph.
What is a necessary and sufficient condition for a bipartite graph.
...
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0answers
151 views
different kind of convergence in Real analysis
Now I am studying real analysis, I wonder if anyone can sum up all the relations between all kinds of convergence, convergence in measure, convergence a.e., convergence a.u., and convergence in ...
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3answers
70 views
If there is only one one-sided limit, the limit exists?
Question on theory: If $\lim_{x\to a^+}F(x)=L$ and $\lim_{x\to a^-}F(x)$ doesn't exist, then does the $\lim_{x\to a}F(x)$ exists and is $L$?
Thanks.
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3answers
197 views
Why does the definition of limits of a function have strict inequality?
Definition (As written in Michael Spivak's Calculus)
The function $f$ approaches a limit $l$ near $a$ means: for every $\epsilon >0$ there is some $\delta > 0$ such that, for all $x$, if ...
3
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1answer
98 views
What is a “set-like class”?
Just / Weese contains the following theorem (p 126):
Theorem 5: Let $\mathbf Z \neq \varnothing$ and let $\mathbf W \subseteq \mathbf Z \times \mathbf Z$ be a wellfounded set-like class. Let $\mathbf ...
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2answers
62 views
Number of Vertices of Graphs
So, I was looking at some graph theoretical stuff, more specifically Topological Graph Theory, and I had a question about the definition of graphs: is there usually a condition in the definition ...
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9answers
2k views
Given real numbers: define integers?
I have only a basic understanding of mathematics, and I was wondering and could not find a satisfying answer to the following:
Integer numbers are just special cases (a subset) of real numbers. ...
0
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0answers
55 views
What is derived number definition? ( in Vitaly covering)
In Vitali covering definition i see "derived number" word, but I dont know what that mean.
Example for vitali covering:
If $f$ is strictly increasing and
$$E=\{x: \text{ there is a derived number } ...
2
votes
0answers
104 views
Explanation of Mixed Strategy Definition in Game Theory
Definition:
Let $(N, A, u)$ be normal-form game, and for any set $X$ let $\Pi(X)$ be the set of all probability distributions over $X$. Then the set of fixed strategies for player $S_i=\Pi(A_i)$.
...
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2answers
2k views
What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?
Ever since I learned the definition of a ring, I've wondered why the additive group is required to be abelian. What happens if we allow $\langle R, + \rangle$ to be nonabelian as well as $\langle R, ...
2
votes
1answer
50 views
Why is the term “composition” used to mean a certain binary operation on the set of relations on a given set?
I looked for material relating to compositions of equivalence relations, and was surprised to find the claim (here) that the composition of equivalence relations is not necessarily again an ...
7
votes
2answers
196 views
True Definition of the Real Numbers
I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a ...
28
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13answers
2k views
What exactly is infinity?
On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and ...
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1answer
65 views
Meaning of index in matrices
Question is, what does "index" mean?
For systems of order greater than the number of characteristic roots of $C$ of index one
Also, can anyone explain why is $u_1 + u_2 + n -1 =0$ and what ...
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0answers
71 views
Definition of stock-flow matrix and understanding it
I am asking a simpler question at here from Constructing and understanding stock-flow model (If one of them needs to be closed, keep this one.)
Suppose that $\textbf{x} = A\textbf{x} + ...
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0answers
71 views
Nothing but a game
I'm trying to study game theory myself but i can't found resources with a formal treating of the theme and from this arise my next questions: what is the formal definition of a game in game theory?
...
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1answer
97 views
What is the difference between a function and a map? [duplicate]
Possible Duplicate:
Is there any difference between mapping and function?
I am an aspiring mathematician who just started out. What is the difference between a function and a map? Or are ...
2
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1answer
118 views
What is the intuitive meaning of the adjugate matrix?
The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything.
What is the intuitive meaning of this matrix?
Are there examples of applications which may ...
3
votes
3answers
60 views
Definition of a tangent
I've been involved in a discussion on definition of a tangent and would appreciate a bit of help.
At my high school and at my college I was taught that a definition of a tangent is 'a line that ...
2
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2answers
159 views
Defining $0^0=1$
I've read in several places that defining $0^0=1$ is convenient in several (primarily discrete) settings. One argument on Wikipedia in favor of this definition was the need of a special case for the ...
